Question: Simplify and expand the following expression: $ \dfrac{3a - 1}{4a + 7}+\dfrac{5a - 3}{a - 6} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4a + 7)(a - 6)$ Multiply the first term by $\dfrac{a - 6}{a - 6}$ $ \begin{align*} \dfrac{3a - 1}{4a + 7} \times \dfrac{a - 6}{a - 6} & = \dfrac{(3a - 1)(a - 6)}{(4a + 7)(a - 6)} \\ & = \dfrac{3a^2 - 19a + 6}{(4a + 7)(a - 6)}\end{align*} $ Multiply the second term by $\dfrac{4a + 7}{4a + 7}$ $ \begin{align*} \dfrac{5a - 3}{a - 6} \times \dfrac{4a + 7}{4a + 7} & = \dfrac{(5a - 3)(4a + 7)}{(a - 6)(4a + 7)} \\ & = \dfrac{20a^2 + 23a - 21}{(a - 6)(4a + 7)}\end{align*} $ Now we have: $ = \dfrac{3a^2 - 19a + 6}{(4a + 7)(a - 6)} + \dfrac{20a^2 + 23a - 21}{(a - 6)(4a + 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3a^2 - 19a + 6 + 20a^2 + 23a - 21}{(4a + 7)(a - 6)} $ $ = \dfrac{23a^2 + 4a - 15}{(4a + 7)(a - 6)}$ Expand the denominator: $ = \dfrac{23a^2 + 4a - 15}{4a^2 - 17a - 42}$